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Comprehending the radio–infrared (IR) relations of the faint extragalactic radio sources is important for using radio emission as a tracer of star formation in high redshift (z) star-forming galaxies (SFGs). Using deep uGMRT observations of the ELAIS-N1 field in the 0.3–0.5 GHz range, we study the statistical properties of the radio–IR relations and the variation of the ‘q-parameter’ up to z = 2 after broadly classifying the faint sources as SFGs and AGN. We find the dust temperature (Tdust) to increase with z. This gives rise to $q_{\rm 24\,\mu m}$, measured at $24\, \mu$m, to increase with z as the peak of IR emission shifts towards shorter wavelengths, resulting in the largest scatter among different measures of q-parameters. $q_{\rm 70\,\mu m}$ measured at $70\, \mu$m, and qTIR using total-IR (TIR) emission are largely unaffected by Tdust. We observe strong, non-linear correlations between the radio luminosities at 0.4 and 1.4 GHz with $70\, \mu$m luminosity and TIR luminosity(LTIR). To assess the possible role of the radio-continuum spectrum in making the relations non-linear, for the first time we study them at high z using integrated radio luminosity (LRC) in the range 0.1–2 GHz. In SFGs, the LRC–LTIR relation remains non-linear with a slope of 1.07 ± 0.02, has a factor of 2 lower scatter compared to monochromatic radio luminosities, and $q^{\rm RC}_{\rm TIR}$ decreases with z as $q^{\rm RC}_{\rm TIR}= (2.27 \pm 0.03)\, (1+z)^{-0.12 \pm 0.03}$. A redshift variation of q is a natural consequence of non-linearity. We suggest that a redshift evolution of magnetic field strengths and/or cosmic ray acceleration efficiency in high-z SFGs could give rise to non-linear radio–IR relations.